Painting with Functions

I recently read (parts of) Paul Lockhart’s 2002 essay about math education. His critique is essentially that we are taking the art out of math by forcing students to focus on every little mechanical subtlety before ever letting them create their own mathematics. Faced with an upcoming unit on polynomials, which I am frankly dreading because I can’t think of a way to make dividing the things interesting, I decided to spend 30 minutes of a class letting the students play at math, to make creations of their own. I started them off with the following ggb file (actually I started them with a set of such files, one with two factors, one with three, one with four, and this one with five). Drag the orange lines around.

I gave them this list of possible questions to get started, but I emphasized that if they found anything fun or interesting, they should feel free to explore that instead, to branch off. Then, for the next 30 minutes, I walked around the classroom admiring what the students were doing, reassuring kids that they really didn’t have to follow any particular instructions, and giving geogebra tips to kids that wanted to move beyond my initial setup.

Results were mixed. At the end of the class period I had some students who were extremely happy with their creations. Some students were upset that they were not able to create anything that they thought was cool. Several students approached real mathematics. One boy noticed that if he put an even number of lines on top of each other, the curve didn’t cross the x-axis, but an odd number of lines would cause the curve to cross, and furthermore that the more lines were on top of each other, the bigger and flatter the flat part would be (e.g. the vertex of a quadratic vs. a quartic). A girl in the corner of the classroom, after changing some of the linear factors to quadratic factors (and later to a sine factor), and changing my original curve from a product of functions to a quotient, noticed that when the dividing function was allowed to reach zero all sorts of crazy stuff happened. Zero students created a formal theorem and proved anything about their observations.

This glimpse of a radically different kind of math education was startling. Some students were really just finger painting, dragging lines around and randomly changing stuff to see what happens. Other kids were trying to create particular effects. Some kids would have continued for another hour, and others were bored and frustrated after 15 minutes. All of this behavior seems a lot like an art class in 1st grade. If these kids had math once or twice a week since elementary school, and it was taught like an art class, do you think they would be up to proving theorems of their own by now? I don’t mean anything revolutionary – I don’t expect we could ever turn every student into a new branch of mathematics – but don’t you think they might be interested in dividing polynomials by 10th grade just to see what happens?

With no time pressure to teach 500 other skills, I would do this every day. Honestly, I was moved by the amount of enjoyment and curiosity I saw in the students, and would scrap everything in the mandatory curriculum after arithmetic in favor of a program like this. I think most students would end up learning MORE math than they do in the lock-stop curriculum through Alg2 / Calculus they go through now.